CORRELATIONS FOR CONVECTIVE HEAT TRANSFER

I. CORRELATIONS FOR FORECD CONVECTION

1. Forced convection from flat plate

Flow regime Range of application Correlation

Laminar, local constwT , 5

xRe 5 10 ,

0.6 Pr 50 1/2 1/3

x xNu 0.322 Re Pr

Laminar, local constwT , 5

xRe 5 10 ,

xRe Pr 100

1/2 1/3x

x 1/42/3

0.3387 Re PrNu

0.04681

Pr

Laminar, local constwq , 5

xRe 5 10 ,

0.6 Pr 50 1/2 1/3

x xNu 0.453 Re Pr

Laminar, local constwq , 5xRe 5 10

1/2 1/3x

x 1/42/3

0.4637 Re PrNu

0.02071

Pr

Laminar, average 5LRe 5 10 , constwT 1/2 1/3

L x=L LNu 2 Nu 0.664 Re Pr

Laminar, local constwT , 5

xRe 5 10 ,

Pr 1 (liquid metals) 1/3 1/3

x xNu 0.564 Re Pr

Laminar, local constwT , starting at 0x x ,

55 10xRe , 0.6 Pr 50

1/33/41/2 1/3 0

x xNu 0.332 Re Pr 1x

x

Turbulent, local constwT , 5 7x5 10 Re 10 2/3 -0.2

x xSt Pr 0.0296 Re

Turbulent, local constwT , 7 9x10 Re 10 2.5842/3

x xSt Pr 0.185 log Re

Turbulent, local constwq , 5 7x5 10 Re 10 x x, =constNu 1.04 Nu

wT

Laminar-turbulent, average

constwT , 7xRe 10 ,

5critRe 5 10

2/3 -0.2 -1L L

1/3 0.8L L

St Pr 0.037 Re 871 Re ,

Nu Pr 0.037 Re 871

Laminar-turbulent, average

constwT , 7xRe 10 , liquids,

at T and w at wT

1/4

0.43 0.8LNu 0.036 Pr 9200

wLRe

High-speed flow constwT , wq hA T T

Same as for low-speed flow with properties evaluated at

* 0.5 0.22w awT T T T T T

[Note: All properties are evaluated at 2f wT T T ]

2. Boundary layer thickness correlations over flat plate

Flow regime Range of application Correlation

Laminar 5xRe 5 10 -1/2

x5.0 Rex

Turbulent 7xRe 10 , 0 at 0x -1/5

x0.381 Rex

Turbulent 5 7

x5 10 < Re 10 , 5

critRe 5 10 , lam at critaRe-1/5 -1x x0.381 Re 10256 Re

x

3. Friction coefficient correlations over flat plate

Flow regime Range of application Correlation

Laminar, local 5xRe 5 10 1/2

x0.332 RefxC

Turbulent, local 5 7x5 10 < Re 10 -1/5

x0.0592 RefxC

Turbulent, local 7 9x10 < Re 10 2.584

x0.37 log RefxC

Turbulent, average 9x crit10 < Re Re 2.854

LL

0.455

Relog Refx

AC

where A can be obtained from the following table

critRe 53 10 55 10 610 63 10 A 1055 1742 3340 8940

4. Nusselt number correlations in forced convection heat transfer over flat plate with unheated starting length

00

x 0 1/3x x 0

0

NuNu , where Nu Re Pr

1

x mxxba

C

x

x

Values of a , b , C and m can be obtained from the following table

Laminar, local Turbulent, local

constwT constwq constwT constwq

a 3/4 3/4 9/10 9/10

b 1/3 1/3 1/9 1/9

C 0.332 0.453 0.0296 0.0308

m 1/2 1/2 4/5 4/5

5. Nusselt number correlations in forced convection heat transfer over sphere and cylinder

Average Nusselt number for forced convection over an isothermal sphere:

1/4

1/2 2/3 0.4D D DNu 2 0.4Re 0.06Re Pr ,

s

which is valid for D3.5 Re 80000 and 0.7 Pr 380 . The fluid properties in this case are

evaluated at the free-stream temperature T , except for s which is evaluated at the surface

temperature sT .

Average Nusselt number for cross flow over an isothermal cylinder:

4/51/2 1/3 5/8D D

D 1/42/3

0.62 Re Pr ReNu 0.3 1 ,

2820000.4

1Pr

which is valid for DRe Pr 0.2 . The fluid properties are evaluated at the film temperature

2f sT T T .

Average Nusselt number for cross flow over isothermal tube banks:

1/4m

D DPr

Nu Re PrPr

n

sC

,

which is valid for 6D0 Re <2 10 and 0.7 Pr 500 . The fluid properties in this case are

evaluated at the free-stream temperature T , except for Prs which is evaluated at

temperature 2m i eT T T , where iT and eT are temperature of the fluid at the inlet and

outlet of the tube bank, respectively.

Typical arrangement of tubes in a tube bank is depicted in the schematic below.

Schematic representation of the arrangement of the tubes in in-line and staggered tube banks (A1, AT, and AD are flow areas at indicated locations, and L is the length of the tubes).

Values of C , m and n can be obtained from the following table (valid when number of tubes is greater than 16)

Arrangement Range of ReD Correlation

In-line

0 100 1/4

0.4 0.36D D

PrNu 0.9 Re Pr

Prs

100 1000 1/4

0.5 0.36D D

PrNu 0.52 Re Pr

Prs

51000 2 10 1/4

0.63 0.36D D

PrNu 0.27 Re Pr

Prs

5 62 10 2 10 1/4

0.8 0.4D D

PrNu 0.033 Re Pr

Prs

Staggered

0 500 1/4

0.4 0.36D D

PrNu 1.04 Re Pr

Prs

500 1000 1/4

0.5 0.36D D

PrNu 0.71 Re Pr

Prs

51000 2 10 1/40.2

0.6 0.36D D

PrNu 0.35 Re Pr

PrT

L s

S

S

5 62 10 2 10 1/40.2

0.8 0.36D D

PrNu 0.031 Re Pr

PrT

L s

S

S

6. Nusselt number correlations in forced convection heat transfer inside circular pipes: case of hydrodynamically and thermally fully developed flow

Laminar flow with isothermal wall: DNu 3.66

Laminar flow with isoflux wall: DNu 4.36

For every other geometry, separate analysis to be made but DNu const .

Turbulent flow:

(i) Dittus-Boelter correlation for smooth wall DRe 10000 : 4/5 nD LNu 0.023 Re Pr (

0.3n for heated wall; 0.4n for cold wall),

(ii) Gnielinski correlation DRe 3000 :

D 1/2 2/3

/ 8 Re 1000 PrNu

1 12.7 / 8 Pr 1

Df

f

( f for

smooth surface can be obtained as 20.790 log Re 1.64Df

, while for rough surface

look into Moody chart)

Above results may be used for other geometries by replacing diameter by hydraulic diameter.

7. Nusselt number correlations in forced convection heat transfer inside circular pipes: Developing region

Laminar flow (isothermal wall):

(i) Combined entry length:

0.141/3Re Pr1.86

/D

Ds

NuL D

for 1/3 0.14Re Pr/ / / 2,D sL D

and 3.66DNu for 1/3 0.14Re Pr/ / / 2D sL D

(ii) Thermal entry length:

2/3

0.0668 / Re Pr3.66

1 0.04 / Re Pr

DD

D

D LNu

D L

Turbulent flow (isothermal wall):

(i) Long tubes with 60L

D : ,D D fdNu Nu

(ii) Short tubes with 60L

D :

,1

/

DmD fd

Nu CNu L D

where 1 and 2 / 3C m

8. Nusselt numbers for fully developed laminar flow in concentric tube annuals

Nusselt number for fully developed laminar flow in a circular tube annulus (of inner diameter

iD and outer diameter oD ) with one surface insulated and the other at constant temperature

Di/Do DiNu DoNu

0 - 3.66 0.05 17.46 4.06 0.1 11.56 4.11

0.258 7.37 4.23 0.5 5.74 4.43 1 4.86 4.86

II. CORRELATIONS FOR NATURAL CONVECTION

1. Nusselt number correlations in natural convection heat transfer over isothermally heated flat plate

Vertical flat plate:

For Pr 0.6 ,

1/41/4

1/4Pr 0.6 : 5 7.07

4x

x

Gr xx xGr

(i) Laminar flow 910LRa :

1/4

4/99/16

0.6700.68

1 0.492 / Pr

LL

RaNu

(ii) Turbulent flow 9 1210 10LRa :

2

1/6

4/99/16

0.3870.825

1 0.492 / Pr

LL

RaNu

Vertical flat plate:

(i) Facing up:

1/40.54L LNu Ra for 4 710 10LRa

1/30.15L LNu Ra for 7 1110 10LRa

(ii) Facing down:

1/40.27L LNu Ra for 5 1010 10LRa

2. Nusselt number correlations in natural convection heat transfer over isothermally heated sphere and cylinder

Sphere:

1/4

4/99/16

0.5892

1 0.469 / Pr

DD

RaNu

Long cylinder:

2

1/612

8/279/16

0.3870.60 for 10

1 0.559 / Pr

DD D

RaNu Ra

Over the years it has been found that average Nusselt number can be represented in the following functional form for a variety of circumstances:

Prm m

f f fNu C Gr C Ra ,

where the subscript f indicates that the properties in the dimensionless groups are evaluated

at the film temperature 2f wT T T .

Values of C and m can be obtained from the following table

Geometry Prf fGr C m

Vertical planes and cylinders

1 410 10 Use Fig. 1 Use Fig. 1

4 910 10 0.59 1/4

9 1310 10 0.021 2/5

9 1310 10 0.1 1/3

Horizontal cylinders

50 10 0.4 0

5 410 10 Use Fig. 2 Use Fig. 2

4 910 10 0.53 1/4

9 1210 10 0.13 1/3

10 210 10 0.675 0.058

2 210 10 1.02 0.148

2 410 10 0.850 0.188

4 710 10 0.480 1/4

7 1210 10 0.125 1/3

Fig. 1:

Fig. 2:

3. Nusselt number correlations in natural convection heat transfer inside enclosures

Average Nusselt number for a horizontal cavity (two of the opposing walls are maintained at

different temperatures 1T and 2T , while other walls are thermally insulated from

surroundings) when heated from below:

1/3 0.074 5 90.069 Pr for 3 10 7 10 ,L LLNu Ra Ra

where all properties are evaluated at the average temperature 1 2 2avT T T .